**
Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 869-878, 2012
**

#
The $M_\alpha$ and $C$-integrals

##
Jae Myung Park, Hyung Won Ryu, Hoe Kyoung Lee, Deuk Ho Lee

* Jae Myung Park, Hyung Won Ryu, Hoe Kyoung Lee*, Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea, e-mail: ` parkjm@cnu.ac.kr`; * Deuk Ho Lee*, Department of Mathematics Education, Kongju National University, Kongju 314-701, South Korea, e-mail: ` dhlee2@kongju.ac.kr`

**Abstract:** In this paper, we define the $M_\alpha$-integral of real-valued functions defined on an interval $[a,b]$ and investigate important properties of the $M_{\alpha}$-integral. In particular, we show that a function $f [a,b]\rightarrow R$ is $M_{\alpha}$-integrable on $[a,b]$ if and only if there exists an $ACG_{\alpha}$ function $F$ such that $F'=f$ almost everywhere on $[a,b]$. It can be seen easily that every McShane integrable function on $[a,b]$ is $M_{\alpha}$-integrable and every $M_{\alpha}$-integrable function on $[a,b]$ is Henstock integrable. In addition, we show that the $M_{\alpha}$-integral is equivalent to the $C$-integral.

**Keywords:** $M_\alpha$-integral, $ACG_\alpha$ function

**Classification (MSC 2010):** 26A39

**Full text** available as PDF.

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade.
To activate your access, please contact Myris Trade at myris@myris.cz.

Subscribers of Springer need to access the articles on their site, which is http://www.springeronline.com/10587.